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{{Trigonometry}}
[[File:Academ Base of trigonometry.svg|thumb|upright=1.35|Basis of trigonometry: if two [[right triangle]]s have equal [[acute angle]]s, they are [[Similarity (geometry)|similar]], so their corresponding side lengths are [[Proportionality (mathematics)|proportional]].]]
In [[mathematics]], the '''trigonometric functions''' (also called '''circular functions''', '''angle functions''' or '''goniometric functions''')
The trigonometric functions most widely used in modern mathematics are the [[sine]], the [[cosine]], and the '''tangent''' functions. Their [[multiplicative inverse|reciprocal]]s are respectively the '''cosecant''', the '''secant''', and the '''cotangent''' functions, which are less used. Each of these six trigonometric functions has a corresponding [[Inverse trigonometric functions|inverse function]], and an analog among the [[hyperbolic functions]].
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== Right-angled triangle definitions ==
[[File:TrigonometryTriangle.svg|thumb|In this right triangle, denoting the measure of angle BAC as A: {{math|1=sin ''A'' = {{sfrac|''a''|''c''}}}}; {{math|1=cos ''A'' = {{sfrac|''b''|''c''}}}}; {{math|1=tan ''A'' = {{sfrac|''a''|''b''}}}}.]]
[[File:TrigFunctionDiagram.svg|thumb|Plot of the six trigonometric functions, the unit circle, and a line for the angle {{math|1=''θ'' = 0.7 radians}}. The points
If the acute angle {{mvar|θ}} is given, then any right triangles that have an angle of {{mvar|θ}} are [[similarity (geometry)|similar]] to each other. This means that the ratio of any two side lengths depends only on {{mvar|θ}}. Thus these six ratios define six functions of {{mvar|θ}}, which are the trigonometric functions. In the following definitions, the [[hypotenuse]] is the length of the side opposite the right angle, ''opposite'' represents the side opposite the given angle {{mvar|θ}}, and ''adjacent'' represents the side between the angle {{mvar|θ}} and the right angle.<ref>{{harvtxt|Protter|Morrey|1970|pp=APP-2, APP-3}}</ref><ref>{{Cite web|title=Sine, Cosine, Tangent|url=https://www.mathsisfun.com/sine-cosine-tangent.html|access-date=29 August 2020|website=www.mathsisfun.com}}</ref>
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[[mnemonics in trigonometry|Various mnemonics]] can be used to remember these definitions.
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, {{math|90°}} or {{math|{{sfrac|π|2}} [[radian]]s}}. Therefore <math>\sin(\theta)</math> and <math>\cos(90^\circ - \theta)</math> represent the same ratio, and thus are equal.
[[File:Periodic sine.svg|thumb|'''Top:''' Trigonometric function {{math|sin ''θ''}} for selected angles {{math|''θ''}}, {{math|{{pi}} − ''θ''}}, {{math|{{pi}} + ''θ''}}, and {{math|2{{pi}} − ''θ''}} in the four quadrants.<br>'''Bottom:''' Graph of sine
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In geometric applications, the argument of a trigonometric function is generally the measure of an [[angle]]. For this purpose, any [[angular unit]] is convenient. One common unit is [[degree (angle)|degrees]], in which a right angle is 90° and a complete turn is 360° (particularly in [[elementary mathematics]]).
However, in [[calculus]] and [[mathematical analysis]], the trigonometric functions are generally regarded more abstractly as functions of [[real number|real]] or [[complex number]]s, rather than angles.
When [[radian]]s (rad) are employed, the angle is given as the length of the [[arc (geometry)|arc]] of the [[unit circle]] subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete [[turn (angle)|turn]] (360°) is an angle of 2{{pi}} (≈ 6.28) rad.
==Unit-circle definitions==
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==Basic identities==
Many [[identity (mathematics)|identities]] interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see [[List of trigonometric identities]]. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval {{math|[0, {{pi}}/2]}}, see [[Proofs of trigonometric identities]]). For non-geometrical proofs using only tools of [[calculus]], one may use directly the differential equations, in a way that is similar to that of the [[#
===Parity===
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{{notelist}}
{{reflist|refs=
<ref name=
<ref name="Larson_2013">{{cite book |title=Trigonometry |edition=9th |first1=Ron |last1=Larson |publisher=Cengage Learning |date=2013 |isbn=978-1-285-60718-4 |page=153 |url=https://books.google.com/books?id=zbgWAAAAQBAJ |url-status=live |archive-url=https://web.archive.org/web/20180215144848/https://books.google.com/books?id=zbgWAAAAQBAJ |archive-date=15 February 2018 }} [https://books.google.com/books?id=zbgWAAAAQBAJ&pg=PA153 Extract of page 153] {{webarchive|url=https://web.archive.org/web/20180215144848/https://books.google.com/books?id=zbgWAAAAQBAJ&pg=PA153 |date=15 February 2018 }}</ref>
<ref name="Aigner_2000">{{cite book |author-last1=Aigner |author-first1=Martin |author1-link=Martin Aigner |author-last2=Ziegler |author-first2=Günter M. |author-link2=Günter Ziegler |title=Proofs from THE BOOK |publisher=[[Springer-Verlag]] |edition=Second |date=2000 |isbn=978-3-642-00855-9 |page=149 |url=https://www.springer.com/mathematics/book/978-3-642-00855-9 |url-status=live |archive-url=https://web.archive.org/web/20140308034453/http://www.springer.com/mathematics/book/978-3-642-00855-9 |archive-date=8 March 2014 }}</ref>
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